IODS week 4: Clustering
Let’s load the data and look at it’s dimensions and structure:
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...## [1] 506 14The data is called Housing Values in Suburbs of Boston and it has information on 14 variables from 506 individual observations. The data is as follows (source):
- crim: per capita crime rate by town.
- zn: proportion of residential land zoned for lots over 25,000 sq.ft.
- indus: proportion of non-retail business acres per town.
- chas: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
- nox: nitrogen oxides concentration (parts per 10 million).
- rm: average number of rooms per dwelling.
- age: proportion of owner-occupied units built prior to 1940.
- dis: weighted mean of distances to five Boston employment centres.
- rad: index of accessibility to radial highways.
- tax: full-value property-tax rate per $10,000.
- ptratio: pupil-teacher ratio by town.
- black: \(1000*(Bk - 0.63)^2\) where Bk is the proportion of blacks by town.
- lstat: lower status of the population (percent).
- medv: median value of owner-occupied homes in $1000s.
Graphical and numerical overview
## -- Attaching packages ---------------------------------- tidyverse 1.2.1 --## v tibble 1.3.4 v purrr 0.2.4
## v tidyr 0.7.2 v dplyr 0.7.4
## v readr 1.1.1 v stringr 1.2.0
## v tibble 1.3.4 v forcats 0.2.0## -- Conflicts ------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()
## x dplyr::select() masks MASS::select()## corrplot 0.84 loaded
## crim zn indus nox
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.3850
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.4490
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.5380
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.5547
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.6240
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :0.8710
## rm age dis tax
## Min. :3.561 Min. : 2.90 Min. : 1.130 Min. :187.0
## 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100 1st Qu.:279.0
## Median :6.208 Median : 77.50 Median : 3.207 Median :330.0
## Mean :6.285 Mean : 68.57 Mean : 3.795 Mean :408.2
## 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188 3rd Qu.:666.0
## Max. :8.780 Max. :100.00 Max. :12.127 Max. :711.0
## ptratio black lstat medv
## Min. :12.60 Min. : 0.32 Min. : 1.73 Min. : 5.00
## 1st Qu.:17.40 1st Qu.:375.38 1st Qu.: 6.95 1st Qu.:17.02
## Median :19.05 Median :391.44 Median :11.36 Median :21.20
## Mean :18.46 Mean :356.67 Mean :12.65 Mean :22.53
## 3rd Qu.:20.20 3rd Qu.:396.23 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :22.00 Max. :396.90 Max. :37.97 Max. :50.00## Charles River dummy variable
## 0 1
## 471 35## Index of accessibility to radial highways
## 1 2 3 4 5 6 7 8 24
## 20 24 38 110 115 26 17 24 132
From these we see that there are some significant correlations between our variables. From the paired plot it is visible that the dependencies are not necessarily linear, rather exponential of nature. Distributions of some are heavily skewed or bimodal (tax or indus).
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865Their means went to zero and variability was also scaled.
## predicted
## correct low med_low med_high high
## low 13 7 1 0
## med_low 4 14 5 0
## med_high 0 14 17 1
## high 0 0 0 25The predictions were quite good. The over all error rate was 0.27 percent.
It seems rad is very influential.
## [1] 405 13## [1] 13 3